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Publication# Approximation for Problems in Multi-User Information Theory

Résumé

The main goal in network information theory is to identify fundamental limits of communication over networks, and design solutions which perform close to such limits. After several decades of effort, many important problems still do not have a characterization of achievable performance in terms of a finite dimensional description. Given this discouraging state of affairs, a natural question to ask is whether there are systematic approaches to make progress on these open questions. Recently, there has been significant progress on several open questions by seeking a (provably) approximate characterization for these open questions. The main goal of approximation in network information theory is to obtain a universal approximation gap between the achievable and the optimal performance. This approach consists of four ingredients: simplify the model, obtain optimal solution for the simplified model, translate this optimal scheme and outer bounds back to the original model, and finally bound the gap between what can be achieved using the obtained technique and the outer bound. Using such an approach, recent progress has been made in several problems such as the Gaussian interference channel, Gaussian relay networks, etc. In this thesis, we demonstrate that this approach is not only successful in problems of transmission over noisy networks, but gives the first approximation for a network data compression problem. We use this methodology to (approximately) resolve problems that have been open for several decades. Not only do we give theoretical characterization, but we also develop new coding schemes that are required to satisfy this approximate optimality property. These ideas could give insights into efficient design of future network communication systems. This thesis is split into two main parts. The first part deals with the approximation in lossy network data compression. Here, a lossy data compression problem is approximated by a lossless counterpart problem, where all the bits in the binary expansion of the source above the required distortion have to be losslessly delivered to the destination. In particular, we study the multiple description (MD) problem, based on the multi-level diversity (MLD) coding problem. The symmetric version of the MLD problem is well-studied, and we can directly use it to approximate the symmetric MD problem. We formulate the asymmetric multi-level diversity problem, and solve it for three-description case. The optimal solution for this problem, which will be later used to approximate the asymmetric multiple description problem, is based on jointly compressing of independent sources. In both symmetric and asymmetric cases, we derive inner and outer bounds for the achievable rate region, which together with the gap analysis, provide an approximate solution for the problem. In particular, we resolve the symmetric Gaussian MD problem, which has been open for three decades, to within 1 bit. In the second part, we initiate a study of a Gaussian relay-interference network, in which relay (helper) nodes are to facilitate competing information flows over a wireless network. We focus on a two-stage relay-interference network where there are weak cross-links, causing the networks to behave like a chain of Z Gaussian channels. For these Gaussian ZZ and ZS networks, we establish an approximate characterization of the rate region. The outer bounds to the capacity region are established using genie-aided techniques that yield bounds sharper than the traditional cut-set outer bounds. For the inner bound of the ZZ network, we propose a new interference management scheme, termed interference neutralization, which is implemented using structured lattice codes. This technique allows for over-the-air interference removal, without the transmitters having complete access to the interfering signals. We use insights gained from an exact characterization of the corresponding linear deterministic version of the problem, in order to study the Gaussian network. We resolve the Gaussian relay-interference network to within 2 bits. The new interference management technique (interference neutralization) shows the use of structured lattice codes in the problem. We also consider communication from a source to a destination over a wireless network with the help of a set of authenticated relays, and presence of an adversarial jammer who wishes to disturb communication. We focus on a special diamond network, and show that use of interference suppression (nulling) is crucial to approach the capacity of the network. The exact capacity characterization for the deterministic network, along with an approximate characterization (to within 4 bits) for the Gaussian network is provided. The common theme that binds the diverse network communication problems in this thesis is that of approximate characterization, when exact resolutions are difficult. The approach of focusing on the deterministic/lossless problems underlying the noisy/lossy network communication problems has allowed us to develop new techniques to study these questions. These new techniques might be of independent interest in other network information theory problems.

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Recent advances in data processing and communication systems have led to a continuous increase in the amount of data communicated over today’s networks. These large volumes of data pose new challenges on the current networking infrastructure that only offers a best effort mechanism for data delivery. The emergence of new distributed network architectures, such as peer-to-peer networks and wireless mesh networks, and the need for efficient data delivery mechanisms have motivated researchers to reconsider the way that information is communicated and processed in the networks. This has given rise to a new research field called network coding. The network coding paradigm departs from the traditional routing principle where information is simply relayed by the network nodes towards the destination, and introduces some intelligence in the network through coding at the intermediate nodes. This in-network data processing has been proved to substantially improve the performance of data delivery systems in terms of throughput and error resilience in networks with high path diversity. Motivated by the promising results in the network coding research, we focus in this thesis on the design of network coding algorithms for simultaneous transmission of multiple data sources in overlay networks. We investigate several problems that arise in the context of inter-session network coding, namely (i) decoding delay minimization in inter-session network coding, (ii) distributed rate allocation for inter-session network coding and (iii) correlation-aware decoding of incomplete network coded data. We start by proposing a novel framework for data delivery from multiple sources to multiple clients in an overlay wireline network, where intermediate nodes employ randomized inter-session network coding. We consider networks with high resource diversity, which creates network coding opportunities with possibly large gains in terms of throughput, delay and error robustness. However, the coding operations in the intermediate nodes must be carefully designed in order to enable efficient data delivery. We look at the problem from the decoding delay perspective and design solutions that lead to a small decoding delay at clients through proper coding and rate allocation. We cast the optimization problem as a rate allocation problem, which seeks for the coding operations that minimize the average decoding delay in the client population. We demonstrate the validity of our algorithm through simulations in representative network topologies. The results show that an effective combination of intra- and inter-session network coding based on randomized linear coding permits to reach small decoding delays and to better exploit the available network resources even in challenging network settings. Next, we design a distributed rate allocation algorithm where the users decide locally how many intra- and inter-session network coded packets should be requested from the parent nodes in order to get minimal decoding delay. The capability to take coding decisions locally with only a partial knowledge of the network statistics is of crucial importance for applications where users are organized in dynamic overlay networks. We propose a receiver-driven communication protocol that operates in two rounds. First, the users request and obtain information regarding the network conditions and packet availability in their local neighborhood. Then, every user independently optimizes the rate allocation among different possible intra- and inter-session packet combinations to be requested from its parents. We also introduce the novel concept of equivalent flows, which permits to efficiently estimate the expected number of packets that are necessary for decoding and hence to simplify the rate allocation process. Experimental results indicate that our algorithm is capable of eliminating the bottlenecks and reducing the decoding delay of users with limited resources. We further investigate the application of the proposed distributed rate allocation algorithm to the transmission of video sequences and validate the performance of our system using the NS-3 simulator. The simulation results show that the proposed rate allocation algorithm is successful in improving the quality of the delivered video compared to intra-session network coding based solutions. Finally, we investigate the problem of decoding the source information from an incomplete set of network coded data with the help of source priors in a finite algebraic field. The inability to form a complete decoding system can be often caused by transmission losses or timing constraints imposed by the application. In this case, exact reconstruction of the source data by conventional algorithms such as Gaussian elimination is not feasible; however, partial recovery of the source data may still be possible, which can be useful in applications where approximate reconstruction is informative. We use the statistical characteristics of the source data in order to perform approximate decoding. We first analyze the performance of a hypothetical maximum a posteriori decoder, which recovers the source data from an incomplete set of network coded data given the joint statistics of the sources. We derive an upper bound on the probability of erroneous source sequence decoding as a function of the system parameters. We then propose a constructive solution to the approximate decoding problem and design an iterative decoding algorithm based on message passing, which jointly considers the network coding and the correlation constraints. We illustrate the performance of our decoding algorithm through extensive simulations on synthetic and real data sets. The results demonstrate that, even by using a simple correlation model expressed as a correlation noise between pairs of sources, the original source data can be partially decoded in practice from an incomplete set of network coded symbols. Overall, this thesis addresses several important issues related to the design of efficient data delivery methods with inter-session network coding. Our novel framework for decoding delay minimization can impact the development of practical inter-session network coding algorithms that are appropriate for applications with low delay requirements. Our rate allocation algorithms are able to exploit the high resource diversity of modern networking systems and represent an effective alternative in the development of distributed communication systems. Finally, our algorithm for data recovery from incomplete network coded data using correlation priors can contribute significantly to the improvement of the delivered data quality and provide new insights towards the design of joint source and network coding algorithms.

The advent of wireless communication technologies has created a paradigm shift in the accessibility of communication. With it has come an increased demand for throughput, a trend that is likely to increase further in the future. A key aspect of these challenges is to develop low complexity algorithms and architectures that can take advantage of the nature of the wireless medium like broadcasting and physical layer cooperation. In this thesis, we consider several problems in the domain of low complexity coding, relaying and scheduling for wireless networks. We formulate the Pliable Index Coding problem that models a server trying to send one or more new messages over a noiseless broadcast channel to a set of clients that already have a subset of messages as side information. We show through theoretical bounds and algorithms, that it is possible to design short length codes, poly-logarithmic in the number of clients, to solve this problem. The length of the codes are exponentially better than those possible in a traditional index coding setup. Next, we consider several aspects of low complexity relaying in half-duplex diamond networks. In such networks, the source transmits information to the destination through $n$ half-duplex intermediate relays arranged in a single layer. The half-duplex nature of the relays implies that they can either be in a listening or transmitting state at any point of time. To achieve high rates, there is an additional complexity of optimizing the schedule (i.e. the relative time fractions) of the relaying states, which can be $2^n$ in number. Using approximate capacity expressions derived from the quantize-map-forward scheme for physical layer cooperation, we show that for networks with $n\leq 6$ relays, the optimal schedule has atmost $n+1$ active states. This is an exponential improvement over the possible $2^n$ active states in a schedule. We also show that it is possible to achieve at least half the capacity of such networks (approximately) by employing simple routing strategies that use only two relays and two scheduling states. These results imply that the complexity of relaying in half-duplex diamond networks can be significantly reduced by using fewer scheduling states or fewer relays without adversely affecting throughput. Both these results assume centralized processing of the channel state information of all the relays. We take the first steps in analyzing the performance of relaying schemes where each relay switches between listening and transmitting states randomly and optimizes their relative fractions using only local channel state information. We show that even with such simple scheduling, we can achieve a significant fraction of the capacity of the network. Next, we look at the dual problem of selecting the subset of relays of a given size that has the highest capacity for a general layered full-duplex relay network. We formulate this as an optimization problem and derive efficient approximation algorithms to solve them. We end the thesis with the design and implementation of a practical relaying scheme called QUILT. In it the relay opportunistically decodes or quantizes its received signal and transmits the resulting sequence in cooperation with the source. To keep the complexity of the system low, we use LDPC codes at the source, interleaving at the relays and belief propagation decoding at the destination. We evaluate our system through testbed experiments over WiFi.

Finding theoretical limits on the performance of communication systems and designing schemes to achieve them is one of the fundamental questions in information theory. While the theory of point-to-point communication is well-investigated, most problems have remained unresolved when communication is over a multi-user system. This lack of understanding is unsatisfactory for today’s growing networks. Recently, significant progress has been made on these problems through a deterministic approach which lays out a promising path in developing a better understanding of multi-user communication systems and in devising new communication schemes. In this thesis, we take a deterministic approach to the problem of communicating nested message sets over wireless and wireline networks. This class of problems is motivated by its applications in video streaming services over heterogeneous networks, where users have different quality-of-service demands. This thesis mainly considers the scenario where a common message (e.g., the low resolution information of a video stream) and a private message (e.g., a higher level of resolution) are to be encoded into a signal and transmitted over a shared medium (e.g., mobile networks, the Internet) towards a set of users. A group of the users, called public receivers, demand only the common message and the rest, called private receivers, demand both messages. The focus is on single-hop broadcast channels and multi-hop wireline networks. A Linear deterministic model for broadcast channels assumes that every user receives a linear transformation of the sent signal. This model is mainly motivated by the MIMO Gaussian broadcast problem in the high SNR regime. We start our study with a simple, yet rich, class of such broadcast channels. We address the main challenges in designing optimal encoding schemes and seek new techniques. In particular, we give an exact characterization of the ultimate rates of communication (together with a class of linear codes that achieves them) over channels with three public and any number of private receivers. We show sub-optimality of these schemes for channels with more than three public receivers and propose a block Markov scheme which allows communication at higher rates. Using this technique, we characterize a set of achievable rates of communication. The intuitions and techniques that we develop over this class of channels guide us towards designing optimal codes for (general) linear deterministic channels. We fully characterize the set of all admissible rates of communication for linear deterministic channels with two public and any number of private receivers. We extend this result to also allow communication of three nested message sets. The study of deterministic models is aimed to be instructive for understanding more general channels. In this regard, we consider the problem of broadcasting two nested message sets over general broadcast channels with two public and one private receiver. For such general broadcast channels, the ultimate rates of communications (and consequently optimal communication schemes) are still unknown. We adapt the block Markov encoding scheme, which we developed within the framework of linear deterministic channels, to general broadcast channels and characterize a set of achievable rates. This achievable rate-region turns out to be equal to the best previously known region (over channels with two public and one private receiver). Nonetheless, we argue that it remains possible that our block Markov encoding scheme may strictly outperform the previous schemes over channels with more (public) receivers. We discuss potential future directions that seem promising. When communication takes place over a multi-hop network, devising optimal communication strategies becomes much more challenging as the encoding scheme of the source should be designed jointly together with the encoding schemes of all the intermediate nodes of the network. We explore this problem over wireline networks, assuming two public and one private receiver. First, we ask if one can always devise a simple and rate-optimal strategy to route the private information to the private receiver and on the remaining network multicast the common message to all receivers. We discuss networks for which this strategy is suboptimal and characterize a class of networks for which it is indeed optimal. We also establish close connections of this problem to linear deterministic channels. This connection lets us formulate another strategy for nodes’ operations which achieves higher rates of transmission. Characterizing all admissible rates of communication over this three-receiver network remains open for further investigation.